LECTURE 10: JACOBI SYMBOL 1. The Jacobi symbol · We wish to generalise the Legendre symbol to accomodate composite p moduli. Definition 1.1. Let Q be an odd positive integer, and suppose that Q = p1 ··· ps, where the pi are prime numbers (not necessarily distinct). Then we a define the Jacobi symbol as follows: Q a (i) = 1; 1 a (ii) = 0 whenever (a; Q) > 1; Q a a a a (iii) = ··· whenever (a; Q) = 1. Q p1 p2 ps Just as in the discussion concerning the Legendre symbol, we begin with some simple properties of the Jacobi symbol. Theorem 1.2. Suppose that Q and Q0 are odd positive integers. Then: P P P (i) = ; Q Q0 QQ0 P P 0 PP 0 (ii) = ; Q Q Q P P 2 (iii) whenever (P; Q) = 1, one has = = 1; Q2 Q P 0P 2 P 0 (iv) whenever (PP 0; QQ0) = 1, one has = ; Q0Q2 Q0 P P 0 (v) whenever P ≡ P 0 (mod Q), one has = . Q Q Proof. Part (i) is immediate from the definition of the Jacobi symbol, and part (ii) is immediate from the properties of the Legendre symbol. Parts (iii) and (iv) follow directly from parts (i) and (ii), since the Jacobi symbol takes values 0 or ±1. For part (v) of the theorem, observe that whenever P ≡ P 0 (mod Q), one has P ≡ P 0 (mod p) for each prime number p dividing Q, whence P P 0 also = for each prime p dividing Q. The desired conclusion is p p therefore again immediate from the definition of the Jacobi symbol. 1 2 LECTURE 10 a Note 1.3. If the Jacobi symbol = −1, then it follows that a is not Q a quadratic residue modulo Q, since for some prime p with p j Q one must a a have that the Legendre symbol = −1. But if = 1, then it is not p Q necessarily the case that a is a quadratic residue modulo Q. For example, one has 2 2 2 = 1; but = −1 and = −1: 15 3 5 The Jacobi symbol remains useful for calculating Legendre symbols, because it satisfies the same reciprocity and simplifying relations as the Legendre sym- bol (as we now demonstrate), and at the same time, whenever the Legendre a symbol is defined (that is, provided that Q is an odd prime number), Q then its value is the same as that of the corresponding Jacobi symbol. Theorem 1.4. Suppose that Q is an odd positive integer. Then −1 2 2 = (−1)(Q−1)=2 and = (−1)(Q −1)=8: Q Q Proof. Suppose that Q is odd, and that Q = p1 : : : ps with each pi a prime number. Then s s −1 Y −1 Y = = (−1)(pi−1)=2: Q p i=1 i i=1 1 But whenever n1 and n2 are both odd, one has 2 (n1 − 1)(n2 − 1) ≡ 0 (mod 2), whence 1 1 1 1 1 2 (n1 −1)+ 2 (n2 −1) = 2 (n1n2 −1)− 2 (n1 −1)(n2 −1) ≡ 2 (n1n2 −1) (mod 2): Iterating the latter relation, we deduce that s 1 X 1 2 (Q − 1) ≡ 2 (pi − 1) (mod 2); i=1 −1 whence = (−1)(Q−1)=2. Q Similarly, we have s s 2 Y 2 Y (p2−1)=8 = = (−1) i : Q p i=1 i i=1 But whenever n1 and n2 are both odd, it follows that 1 2 2 8 (n1 − 1)(n2 − 1) ≡ 0 (mod 2); whence 1 2 1 2 1 2 2 1 2 2 8 (n1 − 1) + 8 (n2 − 1) = 8 (n1n2 − 1) − 8 (n1 − 1)(n2 − 1) 1 2 2 ≡ 8 (n1n2 − 1) (mod 2): LECTURE 10 3 Thus, again iterating this relation, we find that s Q2 − 1 X p2 − 1 ≡ i (mod 2); 8 8 i=1 whence 2 2 = (−1)(Q −1)=8: Q Theorem 1.5 (Quadratic Reciprocity). Suppose that P and Q are odd positive integers with (P; Q) = 1. Then P Q = (−1)(P −1)(Q−1)=4: Q P Proof. Suppose that Q = q1 ··· qs and P = p1 ··· pr are factorisations of P and Q, respectively, into products of prime numbers. Then we have s r s P Y P Y Y pi = = : Q q q j=1 j i=1 j=1 j Then by quadratic reciprocity for the Legendre symbol, we obtain r s P Y Y qj Q = (−1)(pi−1)(qj −1)=4 = (−1)! ; Q p P i=1 j=1 i where we write r s X X (pi − 1)(qj − 1) ! = : 4 i=1 j=1 But as in the proof of Theorem 1.4, one has r s r ! s ! X X (pi − 1)(qj − 1) X pi − 1 X qj − 1 = 4 2 2 i=1 j=1 i=1 j=1 1 1 ≡ 2 (P − 1) · 2 (Q − 1) (mod 2): We therefore deduce that P Q = (−1)(P −1)(Q−1)=4 ; Q P and the conclusion of the theorem now follows immediately. Jacobi symbols are useful for calculating Legendre symbols, since they take the same values for prime moduli, and one can skip intermediate factorisations before applying reciprocity. 1111 Example 1.6. Calculate the Legendre symbol . 8093 4 LECTURE 10 One has 1111 8093 316 2 2 79 = (−1)(1110)(8092)=4 = = 8093 1111 1111 1111 1111 1111 5 79 = (−1)(78)(1110)=4 = − = −(−1)(4)(78)=4 79 79 5 4 22 = − = − = −1: 5 5 So 1111 is not a quadratic residue modulo 8093. Example 1.7. Determine whether or not the congruence x2 + 6x − 50 ≡ 0 (mod 79) has a solution. Observe that x2 +6x−50 = (x+3)2 −59, and hence x2 +6x−50 ≡ 0 (mod 79) 59 has a solution if and only if = 1. But 79 59 −20 −1 2 2 5 5 = = = (−1)(79−1)=2 79 79 79 79 79 79 79 4 = −(−1)(5−1)(79−1)=4 = − = −1: 5 5 Hence the congruence x2 + 6x − 50 ≡ 0 (mod 79) has no solution. Pp x Example 1.8. Let p be an odd prime. Compute x=1 p . Pp x a Let S = x=1 p . There exists a such that p = −1. For instance, this is the case when a is a primitive root modulo p because of Euler's criterion. Since (a; p) = 1, the map x 7! ax (mod p) defines a bijection on the set of residues modulo p. So p p X ax X a x S = = = −S: p p p x=1 x=1 Hence, S = 0. 2. Counting solutions of congruences For an odd prime p and a; b; c 2 Z with (a; p) = 1, we consider the congru- ence y2 ≡ ax2 + bx + c (mod p) (2.1) Let D = b2 − 4ac be the discriminant. Theorem 2.1. The number of solutions with 1 6 x; y 6 p of (2.1) is equal to: a • p − p if p - D, a • p + (p − 1) p if p j D. LECTURE 10 5 Proof. We observe that the number of solutions can be represented as the sum p p X ax2 + bx + c X ax2 + bx + c 1 + = p + : p p x=1 x=1 We observe that 4a(ax2 + bx + c) = (2ax + b)2 − D; Since the map x 7! 2ax + b (mod p) defines a bijection on the set of residues modulo p, we obtain p p X ax2 + bx + c X (4a)−1 (2ax + b)2 − D = p p p x=1 x=1 p (4a)−1 X y2 − D = : p p y=1 By the properties of Legendre symbol, (4a)−1 4a a = = : p p p We write p X y2 − D S(D) = : p y=1 a Then the number of solutions is p + p S(D). When pjD, p X y2 S(D) = = p − 1: p y=1 This immediately implies the second part of the theorem. Suppose then that p - D. We observe that −D X ∗ z − D S(D) = + 2 ; p p 16z6p where the sum is carried out over all non-zero quadratic residues z. We can also rewrite this formula as p X z z − D S(D) = + 1 : p p z=1 6 LECTURE 10 Consider the map z 7! z¯ such thatp ¯ = p andz ¯ satisfies zz¯ ≡ 1 (mod p). Then z z¯ p = p , and p p p X z¯ z − D X zz¯ − Dz¯ X z − D S(D) = + 1 = + p p p p z=1 z=1 z=1 p−1 p X 1 − Dz¯ X z − D = + p p z=1 z=1 p p X 1 − Dz¯ X z − D = −1 + + p p z=1 z=1 Since the map z 7! 1 − Dz¯ (mod p) defines a bijection, we obtain p p X 1 − Dz¯ X x = ; p p z=1 x=1 and similarly, p p X z − D X x = : p p z=1 x=1 Hence, these sums are zero by Example 1.8. This implies that S(D) = −1, which proves the theorem. .